Dear Students, here we shall discuss brief introduction to numbers.
We begin with counting numbers $1,2,3,\ldots$ which are called natural numbers
Integer numbers consist of natural numbers together with their negatives and zero $\ldots,-3,-2,-1,0,1,2,3,\ldots$
We shall use $a,b,\ldots,n$, $p,q,\ldots, x,y, \ldots$ to denote integers, positive or negative.
An integer $a$ is said to be divisible by another integer $b$, not $0$, if there is a third integer $c$ such that $a=bc$.
A rational number is a number that can be expressed as $\dfrac{a}{b}$ in which $a$ is an integer and $b$ is natural number.
The numbers that are not ratios (numbers that are not rational) are called irrational numbers
Real numbers are all "decimal numbers" that correspond to points on number line.
An integer $n$ is called prime number if $n > 1$ and if the only positive divisors of $n$ are $1$ and $n$. If $n > 1$ and if $n$ is not prime, then $n$ is called composite.
Geometric Progression and Exponential Growth:
An arithmetic progression is a list of numbers with the property that to get from any number to the next we need to add a fixed number. In case when addition is replaces by multiplication it is called Geometric progression.
To generate Geometric progression we must be given first number and the constant multiple (known as ratio) that generates successive numbers on the list.
For example we start with $1$ and the ratio is $2$, then geometric progression is: $1,2,4,8,16,32,\ldots$
Note that $\dfrac{2}{1}=2, \dfrac{4}{2}=2, \dfrac{8}{4}=2, \dfrac{16}{8}=2, \dots$ hence the constant multiple is called ratio. It is easy to see that the generic term in this geometric progression is $2^{n}$ for some natural number $n$.
In general if we start with $1$ and have a ratio $r$, then geometric progression is $1,r,r^{2},r^{3},\ldots r^{n},\ldots$. Since the varying quantities is the exponent $n$, we say that $r^{n}$ grows exponentially if $r > 1$ ; for $0 < r < 1$ we say $r^{n}$ decays exponentially
If we start with $1$ and the ratio is $\dfrac{1}{2}$, then geometric progression is: $1,\dfrac{1}{2},\dfrac{1}{4},\dfrac{1}{8},\dfrac{1}{16},\dfrac{1}{32},\ldots$
SET THEORY
Any well-defined collection of objects is a set.If $A$ is a set, then the objects in the collection $A$ are called either the members of $A$ or the elements of $A$.
A set consists of elements (we use numbers) Notation: $x \in M$ means that $x$ is an element of a set $M$ (belongs to $M$).
Some sets occur frequently in mathematics, and it is convenient to adopt a standard notation for them:
$ $$\mathcal{N}$$ $ : the set of all natural numbers (i.e., the numbers 1, 2, 3, etc.)
$ $$\mathcal{Z}$$ $ : the set of all integers (i.e., 0 and all positive and negative whole numbers)
$ $$\mathcal{Q}$$ $: the set of all rational numbers (i.e., fractions)
$ $$\mathcal{R}$$ $: the set of all real numbers
$ $$\mathcal{R+}$$ $ : the set of all non-negative real numbers
There are several ways of specifying a set. If it has a small number of elements we can list them. In this case we denote the set by enclosing the list of the elements in curly brackets; thus, for example, $\{1, 2, 3, 4, 5\}$ denotes the set consisting of the natural numbers 1, 2, 3, 4, and 5. By use of “dots” we can extend this notation to any finite set, e.g., $\{1, 2, 3,\ldots,n\}$ denotes the set of the first $n$ natural numbers. Again $\{2, 3, 5, 7, 11, 13, 17, . . ., 53\}$ could be used to denote the set of all primes up to 53.
Certain infinite sets can also be described by the use of dots (only now the dots have no end), e.g.,
$\{2, 4, 6, 8,\ldots , 2n, \ldots\}$ denotes the set of all even natural numbers. Again, $\{\ldots ,−8,−6,−4,−2, 0, 2, 4, 6, 8, \ldots \}$ denotes the set of all even integers.
In general, however, except for finite sets with only a small number of elements, sets are best described by giving the property which defines the set. If $S(x)$ is some property, the set of all those $x$ that satisfy $S(x)$ is denoted by $\{x \mid S(x)\}$, Or, if we wish to restrict the $x $ to those which are members of a certain set $A$, we would write $\{x \in A \mid S(x)\}$
This is read “the set of all $x$ in $A$ such that $S(x)$”. For example:
$ $$\mathcal{N}$$ = \{x \in $$\mathcal{Z}$$ \mid x > 0\}$
$ $$\mathcal{Q}$$ = \{m/n \mid m, n \in $$\mathcal{Z}$$, n \neq 0 \}$
$\{\sqrt{2}, -\sqrt{2} \} = \{x \in $$\mathcal{R}$$ \mid x^{2} = 2\}$
$\{1, 2, 3\} = \{x \in $$\mathcal{N}$$ \mid x < 4\}$
A set $A$ is a subset of a set $B$ $(A \subset B)$ if each element of $A$ is also an element of $B$. In this case $B$ is called a superset of $A$.
Power Set: For each set there exists a collection of sets that contains among its elements all the subsets of the given set. In other words, if $E$ is a set, then there exists a set (collection) $ $$\mathcal{P}$$ $ such that if $D \subset E$, then $D \in $$\mathcal{P}$$ $
The set $ $$\mathcal{P}$$ = \{D: D \subset E\} $ is called power set of $E$
For example, if $E = \{1, 2, 3\}$, then $ $$\mathcal{P}$$(E) = \{\varnothing, \{1\}, \{2\}, \{3\}, \{1, 2\}, \{1, 3\}, \{2, 3\}, \{1, 2, 3\}\}$
Two sets $A$ and $B$ are equal $(A = B)$ if they consist of the same elements (i.e., if $A \subset B$ and $B \subset A$).
If $A$ is a subset of $B$ and $A \neq B$, then $A$ is called a proper subset of B (notation: $A \subsetneq B$).
The empty set $\varnothing$ (called also the null set) contains no elements. It is a subset of any set.
The empty set can be specified in many
ways, e.g.,
$\varnothing = \{x \in $$\mathcal{R}$$ \mid x^{2} < 0\}$
$\varnothing = \{x \in $$\mathcal{N}$$ \mid 1 < x < 2\}$
$\varnothing = \{x \mid x \neq x\}$
Notice that $\varnothing$ and $ \{\varnothing \}$ are quite different sets. $\varnothing$ is the empty set: it has NO members. $ \{\varnothing \}$ is a set that has ONE member. Hence $\varnothing \neq \{\varnothing \}$
The intersection $A \cap B$ of two sets $A$ and $B$ consists of all elements that belong both to $A$ and to $B$:
$A \cap B = \{x | x \in A$ and $x \in B\}$.
The basic facts about intersections:
$A \cap \varnothing = \varnothing$
$A \cap B = B \cap A$ (commutativity),
$A \cap (B \cap C) = (A \cap B) \cap A$ (associativity),
$A \cap A = A $ (idempotence),
$A \subset B$ if and only if $A \cap B = A$
The union $A \cup B$ consists of all elements of $A$ and all elements of $B$ (and no other elements):
$A \cup B = \{x | x \in A$ or $x \in B\}$.
Here are some facts about the unions of pairs:
$A \cup \varnothing = A$
$A \cup B = B \cup A$ (commutativity),
$A \cup (B \cup C) = (A \cup B) \cup A$ (associativity),
$A \cup A = A $ (idempotence),
$A \subset B$ if and only if $A \cup B = B$
Two useful facts about unions and intersections involve both the operations at the same time:
$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$,
$A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$
These identities are called the distributive laws.
The set difference $A \setminus B$ (or $A-B$, as the relative complement of $B$ in $A$,) consists of elements of $A$ that are not elements of $B$:
$A\setminus B = \{x | x \in A$ and $x \notin B\}$. Note that in this definition it is not necessary to assume that $B \subset A$.
There is a special case: if $B$ is a subset of $A$, the difference $A\setminus B$ is also called a complement of $B$ in $A$.
We assume that all the sets to be mentioned are subsets of one and the same set $U$ and that all complements (unless otherwise specified) are formed relative to that $U$.
An often used symbol for the temporarily absolute (as opposed to relative) complement of $A$ is $A^{\prime}$. In terms of this symbol the basic facts about complementation can be stated as follows:
$(A^{\prime})^{\prime} = A$
$\varnothing^{\prime}=U, U^{\prime} = \varnothing$
$A \cap A^{\prime} = \varnothing, A \cup A^{\prime} = U$
$A \subset B$ if and only if $B^{\prime} \subset A^{\prime}$
The most important statements about complements are the so-called De -Morgan laws:
$(A \cup B)^{\prime} = A^{\prime} \cap B^{\prime}$ and $(A \cap B)^{\prime} = A^{\prime} \cup B^{\prime}$
Here are some easy exercises on complementation.
$A - B = A \cap B^{\prime}$.
$A \subset B$ if and only if $A-B = \varnothing $.
$A- (A-B) = A \cap B$.
$A \cap (B - C) = (A \cap B) - (A \cap C)$.
$A \cap B \subset (A \cap C)\cup (B \cap C^{\prime})$
$(A \cup C)\cap (B \cup C^{\prime}) \subset A \cup B$
The symmetric difference $A\bigtriangleup B$ consists of all elements that belong to exactly one of the sets $A$ and $B$:
$A\bigtriangleup B = (A \setminus B) \cup (B \setminus A) = (A \cup B) \setminus (A \cap B)$.
By $\{a, b, c\}$ we denote the set that contains $a, b, c$ and no other elements. Some of the elements $a, b, c$ may coincide; it this case $\{a, b, c\}$ consists of one or two elements. This notation is also used in a less formal way. For example, the set of all elements of a sequence $a_{1}, a_{2},\ldots, $ is denoted by $\{a_{1}, a_{2},\ldots \}$ (and sometimes even $\{a_{i}\}$). More pedantic notation would be $\{a_{i} \mid i \in $$\mathcal{N}$$ \}$,
Where $ $$\mathcal{N}$$ $ is the set of all natural numbers $($$\mathcal{N}$$ = \{1, 2, . . . \})$
Venn Diagrams:
Real Intervals
By an interval we mean an uninterrupted stretch of the real line. There are a number of different kinds of interval, for which there is a fairly widespread standard notation.
Let $a, b \in $$\mathcal{R}$$, a < b$ The open interval $(a, b)$ is the set $(a, b) = \{x \in $$\mathcal{R}$$ \mid a < x < b\}$
The closed interval $[a, b]$ is the set $[a, b] = \{x \in $$\mathcal{R}$$ \mid a \leq x \leq b \}$
We call $[a, b) = \{x \in $$\mathcal{R}$$ \mid a \leq x < b\}$ a left-closed, right-open interval, and
$(a, b] = \{x \in $$\mathcal{R}$$ \mid a < x \leq b\}$ a left-open, right-closed interval.
Both $[a, b)$ and $(a, b]$ are sometimes referred to as half-open (or half-closed) intervals.
And we set
$(-\infty, a) = \{x \in $$\mathcal{R}$$ \mid x < a\}$
$(-\infty, a] = \{x \in $$\mathcal{R}$$ | x \leq a\}$
$(a,\infty) = \{x \in $$\mathcal{R}$$ | x > a\}$
$[a,\infty) = \{x \in $$\mathcal{R}$$ | x \geq a\}$
We begin with counting numbers $1,2,3,\ldots$ which are called natural numbers
Integer numbers consist of natural numbers together with their negatives and zero $\ldots,-3,-2,-1,0,1,2,3,\ldots$
We shall use $a,b,\ldots,n$, $p,q,\ldots, x,y, \ldots$ to denote integers, positive or negative.
An integer $a$ is said to be divisible by another integer $b$, not $0$, if there is a third integer $c$ such that $a=bc$.
A rational number is a number that can be expressed as $\dfrac{a}{b}$ in which $a$ is an integer and $b$ is natural number.
The numbers that are not ratios (numbers that are not rational) are called irrational numbers
Real numbers are all "decimal numbers" that correspond to points on number line.
An integer $n$ is called prime number if $n > 1$ and if the only positive divisors of $n$ are $1$ and $n$. If $n > 1$ and if $n$ is not prime, then $n$ is called composite.
Geometric Progression and Exponential Growth:
An arithmetic progression is a list of numbers with the property that to get from any number to the next we need to add a fixed number. In case when addition is replaces by multiplication it is called Geometric progression.
To generate Geometric progression we must be given first number and the constant multiple (known as ratio) that generates successive numbers on the list.
For example we start with $1$ and the ratio is $2$, then geometric progression is: $1,2,4,8,16,32,\ldots$
Note that $\dfrac{2}{1}=2, \dfrac{4}{2}=2, \dfrac{8}{4}=2, \dfrac{16}{8}=2, \dots$ hence the constant multiple is called ratio. It is easy to see that the generic term in this geometric progression is $2^{n}$ for some natural number $n$.
In general if we start with $1$ and have a ratio $r$, then geometric progression is $1,r,r^{2},r^{3},\ldots r^{n},\ldots$. Since the varying quantities is the exponent $n$, we say that $r^{n}$ grows exponentially if $r > 1$ ; for $0 < r < 1$ we say $r^{n}$ decays exponentially
If we start with $1$ and the ratio is $\dfrac{1}{2}$, then geometric progression is: $1,\dfrac{1}{2},\dfrac{1}{4},\dfrac{1}{8},\dfrac{1}{16},\dfrac{1}{32},\ldots$
SET THEORY
Any well-defined collection of objects is a set.If $A$ is a set, then the objects in the collection $A$ are called either the members of $A$ or the elements of $A$.
A set consists of elements (we use numbers) Notation: $x \in M$ means that $x$ is an element of a set $M$ (belongs to $M$).
Some sets occur frequently in mathematics, and it is convenient to adopt a standard notation for them:
$ $$\mathcal{N}$$ $ : the set of all natural numbers (i.e., the numbers 1, 2, 3, etc.)
$ $$\mathcal{Z}$$ $ : the set of all integers (i.e., 0 and all positive and negative whole numbers)
$ $$\mathcal{Q}$$ $: the set of all rational numbers (i.e., fractions)
$ $$\mathcal{R}$$ $: the set of all real numbers
$ $$\mathcal{R+}$$ $ : the set of all non-negative real numbers
There are several ways of specifying a set. If it has a small number of elements we can list them. In this case we denote the set by enclosing the list of the elements in curly brackets; thus, for example, $\{1, 2, 3, 4, 5\}$ denotes the set consisting of the natural numbers 1, 2, 3, 4, and 5. By use of “dots” we can extend this notation to any finite set, e.g., $\{1, 2, 3,\ldots,n\}$ denotes the set of the first $n$ natural numbers. Again $\{2, 3, 5, 7, 11, 13, 17, . . ., 53\}$ could be used to denote the set of all primes up to 53.
Certain infinite sets can also be described by the use of dots (only now the dots have no end), e.g.,
$\{2, 4, 6, 8,\ldots , 2n, \ldots\}$ denotes the set of all even natural numbers. Again, $\{\ldots ,−8,−6,−4,−2, 0, 2, 4, 6, 8, \ldots \}$ denotes the set of all even integers.
In general, however, except for finite sets with only a small number of elements, sets are best described by giving the property which defines the set. If $S(x)$ is some property, the set of all those $x$ that satisfy $S(x)$ is denoted by $\{x \mid S(x)\}$, Or, if we wish to restrict the $x $ to those which are members of a certain set $A$, we would write $\{x \in A \mid S(x)\}$
This is read “the set of all $x$ in $A$ such that $S(x)$”. For example:
$ $$\mathcal{N}$$ = \{x \in $$\mathcal{Z}$$ \mid x > 0\}$
$ $$\mathcal{Q}$$ = \{m/n \mid m, n \in $$\mathcal{Z}$$, n \neq 0 \}$
$\{\sqrt{2}, -\sqrt{2} \} = \{x \in $$\mathcal{R}$$ \mid x^{2} = 2\}$
$\{1, 2, 3\} = \{x \in $$\mathcal{N}$$ \mid x < 4\}$
A set $A$ is a subset of a set $B$ $(A \subset B)$ if each element of $A$ is also an element of $B$. In this case $B$ is called a superset of $A$.
Power Set: For each set there exists a collection of sets that contains among its elements all the subsets of the given set. In other words, if $E$ is a set, then there exists a set (collection) $ $$\mathcal{P}$$ $ such that if $D \subset E$, then $D \in $$\mathcal{P}$$ $
The set $ $$\mathcal{P}$$ = \{D: D \subset E\} $ is called power set of $E$
For example, if $E = \{1, 2, 3\}$, then $ $$\mathcal{P}$$(E) = \{\varnothing, \{1\}, \{2\}, \{3\}, \{1, 2\}, \{1, 3\}, \{2, 3\}, \{1, 2, 3\}\}$
Two sets $A$ and $B$ are equal $(A = B)$ if they consist of the same elements (i.e., if $A \subset B$ and $B \subset A$).
If $A$ is a subset of $B$ and $A \neq B$, then $A$ is called a proper subset of B (notation: $A \subsetneq B$).
The empty set $\varnothing$ (called also the null set) contains no elements. It is a subset of any set.
The empty set can be specified in many
ways, e.g.,
$\varnothing = \{x \in $$\mathcal{R}$$ \mid x^{2} < 0\}$
$\varnothing = \{x \in $$\mathcal{N}$$ \mid 1 < x < 2\}$
$\varnothing = \{x \mid x \neq x\}$
Notice that $\varnothing$ and $ \{\varnothing \}$ are quite different sets. $\varnothing$ is the empty set: it has NO members. $ \{\varnothing \}$ is a set that has ONE member. Hence $\varnothing \neq \{\varnothing \}$
The intersection $A \cap B$ of two sets $A$ and $B$ consists of all elements that belong both to $A$ and to $B$:
$A \cap B = \{x | x \in A$ and $x \in B\}$.
The basic facts about intersections:
$A \cap \varnothing = \varnothing$
$A \cap B = B \cap A$ (commutativity),
$A \cap (B \cap C) = (A \cap B) \cap A$ (associativity),
$A \cap A = A $ (idempotence),
$A \subset B$ if and only if $A \cap B = A$
The union $A \cup B$ consists of all elements of $A$ and all elements of $B$ (and no other elements):
$A \cup B = \{x | x \in A$ or $x \in B\}$.
Here are some facts about the unions of pairs:
$A \cup \varnothing = A$
$A \cup B = B \cup A$ (commutativity),
$A \cup (B \cup C) = (A \cup B) \cup A$ (associativity),
$A \cup A = A $ (idempotence),
$A \subset B$ if and only if $A \cup B = B$
Two useful facts about unions and intersections involve both the operations at the same time:
$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$,
$A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$
These identities are called the distributive laws.
The set difference $A \setminus B$ (or $A-B$, as the relative complement of $B$ in $A$,) consists of elements of $A$ that are not elements of $B$:
$A\setminus B = \{x | x \in A$ and $x \notin B\}$. Note that in this definition it is not necessary to assume that $B \subset A$.
There is a special case: if $B$ is a subset of $A$, the difference $A\setminus B$ is also called a complement of $B$ in $A$.
We assume that all the sets to be mentioned are subsets of one and the same set $U$ and that all complements (unless otherwise specified) are formed relative to that $U$.
An often used symbol for the temporarily absolute (as opposed to relative) complement of $A$ is $A^{\prime}$. In terms of this symbol the basic facts about complementation can be stated as follows:
$(A^{\prime})^{\prime} = A$
$\varnothing^{\prime}=U, U^{\prime} = \varnothing$
$A \cap A^{\prime} = \varnothing, A \cup A^{\prime} = U$
$A \subset B$ if and only if $B^{\prime} \subset A^{\prime}$
The most important statements about complements are the so-called De -Morgan laws:
$(A \cup B)^{\prime} = A^{\prime} \cap B^{\prime}$ and $(A \cap B)^{\prime} = A^{\prime} \cup B^{\prime}$
Here are some easy exercises on complementation.
$A - B = A \cap B^{\prime}$.
$A \subset B$ if and only if $A-B = \varnothing $.
$A- (A-B) = A \cap B$.
$A \cap (B - C) = (A \cap B) - (A \cap C)$.
$A \cap B \subset (A \cap C)\cup (B \cap C^{\prime})$
$(A \cup C)\cap (B \cup C^{\prime}) \subset A \cup B$
The symmetric difference $A\bigtriangleup B$ consists of all elements that belong to exactly one of the sets $A$ and $B$:
$A\bigtriangleup B = (A \setminus B) \cup (B \setminus A) = (A \cup B) \setminus (A \cap B)$.
By $\{a, b, c\}$ we denote the set that contains $a, b, c$ and no other elements. Some of the elements $a, b, c$ may coincide; it this case $\{a, b, c\}$ consists of one or two elements. This notation is also used in a less formal way. For example, the set of all elements of a sequence $a_{1}, a_{2},\ldots, $ is denoted by $\{a_{1}, a_{2},\ldots \}$ (and sometimes even $\{a_{i}\}$). More pedantic notation would be $\{a_{i} \mid i \in $$\mathcal{N}$$ \}$,
Where $ $$\mathcal{N}$$ $ is the set of all natural numbers $($$\mathcal{N}$$ = \{1, 2, . . . \})$
Venn Diagrams:
Real Intervals
By an interval we mean an uninterrupted stretch of the real line. There are a number of different kinds of interval, for which there is a fairly widespread standard notation.
Let $a, b \in $$\mathcal{R}$$, a < b$ The open interval $(a, b)$ is the set $(a, b) = \{x \in $$\mathcal{R}$$ \mid a < x < b\}$
The closed interval $[a, b]$ is the set $[a, b] = \{x \in $$\mathcal{R}$$ \mid a \leq x \leq b \}$
We call $[a, b) = \{x \in $$\mathcal{R}$$ \mid a \leq x < b\}$ a left-closed, right-open interval, and
$(a, b] = \{x \in $$\mathcal{R}$$ \mid a < x \leq b\}$ a left-open, right-closed interval.
Both $[a, b)$ and $(a, b]$ are sometimes referred to as half-open (or half-closed) intervals.
And we set
$(-\infty, a) = \{x \in $$\mathcal{R}$$ \mid x < a\}$
$(-\infty, a] = \{x \in $$\mathcal{R}$$ | x \leq a\}$
$(a,\infty) = \{x \in $$\mathcal{R}$$ | x > a\}$
$[a,\infty) = \{x \in $$\mathcal{R}$$ | x \geq a\}$
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